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III. Potensrækker – 4. Taylorpolynomier 308 4.19. Taylors restled ☞ [S] 8.7 Taylor and Maclaurin series x = f(a) + (f ′ s (a) + f ′′ (t) dt) ds a = f(a) + (x − a)f ′ x s (a) + f ′′ (t) dt ds. 4.20. Taylors restled ☞ [S] 8.7 Taylor and Maclaurin series f(x) = f(a) + (x − a)f ′ x s (a) + f ′′ (t) dt ds. De to første led er Taylor-polynomiet T1(x) for f, og det sidste led er derfor en formel for restleddet R1(x). 4.21. Taylors restled ☞ [S] 8.7 Taylor and Maclaurin series Vi kan genkende dette itererede integral som et udtryk for (plus/minus) dobbeltintegralet af f ′′ (t) over trekanten D i (s,t)-planen, afgrænset af t = a (vandret linie), s = x (lodret linie) og linien s = t Trekanten D har areal 1 2! (x − a)2 . Da |f ′′ (t)| ≤ M for alle punkter i D, er D ≤ 1 2! (x − a)2 · M = M (x − a)2 2! a a a a a
III. Potensrækker – 4. Taylorpolynomier 309 4.22. Eksponentialrækken konvergerer mod eksponentialfunktionen ☞ [S] 8.7 Taylor ... Eksempel 2 Tn(x) = 1 + x x2 xn + + ... + 1! 2! n! er afsnits-summen i eksponentialrækken. For hvilke x gælder For hvilke x gælder For alle x ! THI: Tn(x) → e x for n → ∞? Rn(x) → 0 for n → ∞? 4.23. Eksponentialrækken ☞ [S] 8.7 Taylor and Maclaurin series tag et d ≥ x. I intervallet [−d,d] er så restledsvurderingen giver f (n+1) (x) = e x ≤ e d |Rn(x)| ≤ for |x| ≤ d. Men |x|n+1 (n+1)! → 0 for n → ∞. Altså Rn(x) → 0 for n → ∞ Altså Tn(x) → f(x) = e x for n → ∞. e d (n + 1)! |x|n+1
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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3. Generelle områder 201 4. Koordi
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